We consider spatially periodic growth models built from weakly asymmetric exclusion processes with finite-range jumps and rates depending locally on configuration of particles. We prove that at a large scale and after renormalization these processes converge to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. Since the driving noise of the discrete equation decorrelates slowly in time, the hydrodynamic behaviour of the system needs to be exploited.